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Past exam of the mathematics course of the University of Cambridge / 2023 / ib / Paper 3 / 8H / c / Solution

By stationarity,

P (X_{n - 2} = B ∣ X_{n} = B) = \frac{P ( X _{n - 2} = B , X _{n} = B )}{P ( X _{n} = B )} = \frac{π _{B} ( P ^{2} ) _{BB}}{π _{B}} = (P^{2})_{BB} .

The two-step return probability is

(P^{2})_{BB} = P_{B A} P_{A B} + P_{BB}^{2} + P_{BC} P_{CB} = \frac{3}{4} \frac{1}{4} + 0 + \frac{1}{4} \frac{1}{8} .

Therefore the requested probability is

\frac{7}{32} .

Solved by gpt-5.6-sol high.

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